1992 - X11 Symmetric Linear Filter & Their Transfer Functions - Bell, Monsel

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BUREAU OF THE CENSUS

STATISTICAL RESEARCH DIVISION

RESEARCH REPORT SERIES

No. RR-92/15

X-11 Symmetric Linear Filters and

their Transfer Functions

bY

William R. Bell

and

Brian C. Monsell

Bureau of the Census

Washington, DC 20233-42--

Report Issued: 12/10/92

This series contains research reports, written by or in cooperation with, staff members of the

Statistical Research Division, whose content may be of interest to the general statistical research

community. The views reflected in these reports are not necessarily those of the Census Bureau

nor do they necessarily represent Census Bureau statistical policy or practice. Inquiries may be



1. Introduction

The X-11 program (Shiskin, Young, and Musgrave 1967) is widely used for seasonal

adjustment of economic time series. One can view additive X-11 as a linear filtering

operation produced by successive application of simple linear filters - seasonal and

nonseasonal moving averages and Henderson trend moving averages. In the additive

decomposition this view is exact except for modifications made to deal with extreme values.

This same linear filter view applies to the log-additive decomposition of X-ll-ARIMA

(Dagum 1983), except that it applies to the logarithms of the original time series.

In the

multiplicative decomposition the linear filter view differs from reality in that results obtained

from applying a moving average at any stage are then divided into the series at hand, rather

*

than subtracted from it, to produce results for the next stage (hence the name “ratio to

motig average method”). Young (1968), however, assessed the differences between a linear

approximation to and actual multiplicative X-11, and argued that the differences due to

nonlinearities are generally unimportant. To this degree of approximation, therefore, the

X-11 linear filters are also relevant to the multiplicative decomposition.

This note presents graphs of titer weights and transfer function squared gains

corresponding to symmetric linear X-11 seasonal, seasonal adjustment, trend, and irregular

filters, under the various choices of fiIter options allowed. We focus exclusively on the

symmetric titers for two reasons: (1) the practice of revising initial seasonal adjustments to

the final adjustments obtained from the symmetric filters suggests that these filters should be

of prime interest; and (2) applying the symmetric fiIters to a series extended with as many

minimum mean squared error forecasts and backcasts as needed (in the spirit of

X-ll-ARIMA (Dagum 1983))

minimizes mean squared seasonal adjustment revisions, as was

shown by Geweke (1978) and Pierce (1980). If desired, one could obtain representations of



the symmetric and asymmetric moving averages. We do not pursue this here, as the possible

number of asymmetric filters under the various options would be quite large.

2. Commutation of X-11 Svmmetric Linear Filters and Their Transfer Functions

Young (1968) and Wallis (1974) have offered linear filter approximations to X-11. We

follow the approach of WaIlis, whose approximation to X-11 is the more complete.

(As noted

by WaIlis, Young omits two steps from X-11.) Let Zt = St + Tt + It be an additive, or

log-additive, decomposition of an observed time series Zt as seasonal + trend + irregular.

Henceforth, let “MA” denote “moving average”.

Wallis (1974) lists the filtering steps used

e by X-11 in estimating this decomposition, a summary description of which is as follows:

-1.)

2.1

3.1

4-l

5.1

6.1

7.1

8-l

Detrend Zt by subtracting a 2x12 MA (a 2-term MA of a la-term MA).

Take a first seasonal MA (default =

3x3) of the result as a preliminary estimate

of St.

Adjust the preliminary seasonal to sum more nearly to 0 over 12 months by

subtracting a 2x12 MA.

Subtract the result of 3. from Zt to get a preliminary seasonally adjusted series.

Subtract a Henderson trend MA of this from Zt for a more refined detrending.

Apply a second seasonal MA (default = 3x5) to the result of 5.

Adjust the result of 6. as in 3. by subtracting a 2x12 MA - the result, St,

estimates St.

The seasonally adjusted series (nonseasonal estimate) is then fit = Zt - St, the

trend estimate, T,, is obtained by applying a Henderson trend MA to fit, and the



3

variation. Thus, the above gives the basic filtering steps of (additive) X-11.

We can express each of the above steps in terms of polynomials in the backshift

operator, B (BZt

= Zt-1), and then write down the resulting backshift operator polynomial

representations of the X-11 seasonal, seasonal adjustment, trend, and irregular filters. Doing

this produces the following expression for the X-11 seasonal fdter, us(B):

+) = [l - 3)1X2(B)[1H(B){1 [l - ~ )l~~(B)[l- /@)I)] (2.1)

where, letting U(B) = l+B + . . + B1’ and F = B-l, we write

= 2x12 trend moving average

= (1/24)(F6 + F5)U(B)

= first seasonal moving average, the default is the 3x3 seasonal MA:

(1/9)(F12 + 1 + B12)(F12 + 1 + B12)

= second seasonal moving average, the default is the 3x5 seasonal MA:

(l/15)(F12 + 1 + B12)(F24 + F12 + 1 + B12 + B24)

= Henderson trend moving average.

For the quarterly titer change 12 to 4 and 24 to 8 in the above expressions.

The seasonal

adjustment (%(B)), trend (wT(B)),

and irregular (q(B)) filters are obtained from the

seasonal filter, us(B), as follows:

*(B) = 1- q3)

(2.2)



The expressions (2.1) - (2.4) provide a convenient means of computing the weights and

transfer functions for the various filters resulting from selection of default or optional choices

of Xl(B), X2(B), and H(B). If w(B) = k %Bk is an X-11 filter (seasonal, adjustment,

trend, or irregular), the filter weights wk can then be computed using a mathematical

programming language that will multiply polynomials; GAUSS (Edlefsen and Jones 1986)

was used here. The transfer function of u(B) is then de2”“) = E %e2fiXk for

X E [-.5, .5]. For symmetric w(B), i.e. wk = u-k, it turns out that de2tix) is real valued

with w(e2dX) = ,(es2”“) and squared gain [,(e2”“)12. Thus, for the symmetric filters,

* we only need plot the weights wk for k 2 0, and we only need plot [,(e2”‘ 12 for X E [0, .5].

*The expressions (2.1) to (2.4) also facilitate study of the properties of the X-11 filters.

This is pursued in Bell (1992).

The optional filters we present here are those available in the version of X-11 currently

being distributed by the Census Bureau, the X-11.2 program (Monsell 1989). The default

seasonal MA choices are as given above; optional 3-term, 3x3, 3x5, or 3x9 seasonal MAs can

also be chosen. The choice of an optional seasonal MA implies that it is used for both Xl(B)

and X2(B). A Vixed seasonal filter” with equal weights can also be used, but as this is more

akin to regression on seasonal dummies, and since the weights obtained under this filter

depend on the length of the series, it shall also not be considered here.

For monthly series the X-11.2 user can optionally select a 9-term, 13-term, or 23-term

Henderson trend MA, and for quarterly series a 5-term or ‘I-term Henderson trend MA.

Weights for the n-term Henderson MA, H,(B) = C htn)Bj, are hfn), j = O,fl,...,f(n-1)/2

j ’

obtained from the following formula, with m

= (n+3)/2, given by Macaulay (1931, p. 54) and



5

h(n) 315[(m-1) 2- j 2][m2- j 2][(m+ 1 )2-j2] [ (3m2-16)-llj2]

=

J

8m(m2-1 ) (4m2-1) ( 4m2-9)(4m2-2 5)

(2.5)

.

The default option is for the X-11 program to automatically choose one of the available

Henderson trend MAs, with the choice depending on the relative amounts of variation in

preliminary estimates of the irregular and trend.

We made a quick check of 30 Census

Bureau monthly time series and found that X-11 picked the 9-term, 13-term, and 23-term

flters 10, 18, and 2 times, respectively. It is possible for the automatic choice of H(B) used

in (2.3) and (2.4) to differ from that used in (2.1), since the Henderson trend filter used in

.

(2.3) and (2.4) is determined independently, though in the same fashion as for (2.1). For the

30 series mentioned earlier, a different choice of H(B) in (2.1) versus (2.3)-(2.4) was made

only bnce. To avoid a large increase in the number of graphs required, while probably risking

little loss of relevant information, we present graphs only for the case where H(B) is the same

in (2.3) and (2.4) as it is in (2.1).

The options provided by X-11.2 differ from the other well-known incarnations of X-11:

the original X-11 program described by Shiskin, Young, and Musgrave (1967); and the

X-11-AR.IMA program (Dagum 1983). X-ll-ARIMA drops the optional 3-term seasonal

MA, but also allows an optional 2x24 MA (monthly) or 2x8 MA (quarterly) in place of Ir(B).

The original X-11 program does not provide optional seasonal or Henderson trend MAs for

quarterly series. (The 5-term Henderson is always used.) Original X-11 also differs from

X-11.2 and X-ll-ARIMA in one other arbitrary way, in that it stores the various MAs used

to only 3 decimals, whereas X-11.2 and X-ll-ARIMA use more accurate representations.

This difference should not materially affect the graphs presented here.



6

weights, followed by the graphs of all the squared gains of the monthly filters. This pattern

is then repeated for all the quarterly filters. Within each of these sets, the graphs are

grouped in the order of seasonal, adjustment, trend, and irregular filters. For each of these

groups of graphs, there is one page of graphs for each choice of Henderson trend MA, in the

order of %term, C&term, and 23-term for the monthly graphs, and 5-term and ‘I-term for

the quarterly graphs. Each page contains four graphs corresponding to alternative choices of

seasonal MAs: default, 3x3, 3x9, and 3-term, in this order, clockwise from upper left. This

ordering is given in abbreviated outline form on pages 7 and 8.

Upon examining graphs for the “optional” 3x5 seasonal MA we discovered that these

+ were virtually identical to those for the default seasonal MA. Thus, the graphs presented for

the default seasonal MA can also serve as graphs for the optional 3x5. Another way to look

at &a is that the 3x5 is not really a distinct option from the default seasonal MA, at least as

far as the symmetric filter is concerned. (We have not compared asymmetric filters resulting

from the default and optional 3x5 seasonal MAs.)

A few of the graphs of filter weights and gains that follow have been previously

produced by other authors. Wallis (1974) plotted weights for the monthly and quarterly

adjustment filters for the default seasonal MA and K&term Henderson trend MA @-term for

the quarterly filter). He also graphed the squared gain of the monthly adjustment filter.

Some of these plots are repeated in Wallis (1982) and Burridge and Wallis (1984), along with

plots for some asymmetric filters. Young (1968) plotted monthly seasonal, trend, and

irregular filter weights from his approximation to X-11 for the default seasonal MA with

9-term, K&term, and 23-term Henderson filters. He also plotted end weights for the seasonal

and trend filters. Those plots of Young and Wallis that correspond to cases treated here do



7

Organimtion of Graphe - Outline

1. Monthly Filter Weights

1.1 Seasonal Filters

1.1.1 9-term Henderson Trend MA

1.1.1.1 Default Seasonal MA

1.1.1.2 Optional 3x3 Seasonal MA


1.1.1.4 Optional 3-term Seasonal MA

1.1.2 U-term Henderson Trend MA



1.2 Adjustment Filters

.

.

.

1.3 Trend Filters

1.4 Irregular Filters

.

2. Monthly Filter Transfer Functions



2.3 Trend Filters

Squared Gains

.

.




8

3. Quarterly Filter Weights






3.1.1.4 Optional 3-term Seasonal MA

3.1.2 ‘I-term Henderson Trend MA


.

.

.


3.3 Trend Filters

.

*


.

.

4. Quarterly Filter Transfer Functions - Squared Gains


.


.

4.3 Trend Filters


.



9

REFERENCES

Bell, W. R. 1992) “On Some Properties of X-11 Symmetric Linear Filters,” Research

Report, tatisticai Research Division, U. S. Bureau of the Census, forthcoming.

Burridge, P. and Wallis, K. F. (1984) “Unobserved Components Models for Seasonal

Adjustment Filters,”

Journal of Business and Economic Statistics, 2, 350-359.

Dagum, E. B. (1983), “The X-ll-ARIMA Seasonal Adjustment Method,” Catalogue

12-5643, Time Series Research and Analysis Staff, Statistics Canada.

(1985) “Moving Averages,” in Encvclonedia of Statistical Sciences, Vol. 5, ed. S.

Kotz and N. L. Johnson, New York: John Wiley, 639-634.

Ed&en, L. E. and Jones, S. D. (1986) GAUSS Programming Language Manual, Seattle:

Aptech Systems, Inc.

Geweke, J. (1978) “Revision of Seasonally Adjusted Time Series,” SSRI Report No. 7822,

Department of Economics, University of Wisconsin.

* Macaulay, F. R. (1931) The Smoothing of Time Series, New York: National Bureau of

Economic Research.

Monaell, B. C. (1989) “Supplement to Census Technical Paper No. 15: The Uses and

Features of X-11.2 and X-llQ.2,” Time Series Staff, Statistical Research Division,

U. S. Bureau of the Census.

Pierce, D. A. 1980 ,

c )

“Data Revisions With Moving Average Seasonal Adjustment

Procedures,’ Journal of Econometrics, 14, 95-114.

Shiskin, J., Young, A. H., and Musgrave, J. C.

Method II Seasonal Adjustment Program,”

I

1967), “The X-11 Variant of the Census

of Commerce, Bureau of Economic Analysis.

echnicai Paper No. 15, U.S. Department

Wallis, K. F. (1974 ,

the American

d

“Seasonjl Adjustment and Relations between Variables,” Journal of

tatistical Association, 69, 18-31.

(1982), “Seasonal Adjustment and Revision of Current Data: Linear Filters for

the X-11 Method,” Journal of the Royal Statistical Societv, Ser. A, 145, 74-85.

Wolter, K. M. and Monsour, N. J. (1981), “On the Problem of Variance Estimation for a

Deseasonalixed Series,” in Current Tonics in Survev Samnlinq, ed. D. Krewski, R.

Platek, and J.N.K. Rao, New York: Academic Press, 199-226.

Young, A. H. (1968), “Linear Approximations to the Census and BLS Seasonal

Adjustment Methods,” Journal of the American Statistical Association, 63, 445-471.



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1992 - X11 Symmetric Linear Filter & Their Transfer Functions - Bell, Monsel

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